e-Statistics

## Analysis of Variance

A model introduces the population mean for each level . Here the parameter is the overall average, and is known as the i-th factor effect. The hypothesis testing problem to detect some effects'' of factor level becomes

versus     for some .

Equivalently we can write the hypothesis testing problem as follows:

versus     for some .

The data from k groups are arranged either (a) all in a single variable with another categorical variable indicating factor levels, or (b) in multiple columns each of whose variables represents a factor level. The statistical inference begins with calculation of the sample mean within group for every factor level , which is the point estimate of . It is also useful to obtain the sample standard deviation within the group, that is, the square root of .

We then proceed to compute the analysis of variance table (AOV table) which summarizes the degree of freedom (df), the sum of squares (SS), and mean squares (MS).

1. is the sum of squares between groups, having degrees of freedom. Thus, the mean sqaure is given by

2. is the sum of squares within groups, having degrees of freedom. Thus, the mean sqaure is given by

3. is the total sum of squares, having degrees of freedom. It can be decomposed into

Here

is the overall sample mean with the total sample size , and represents the point estimate of . The statistical model assumes (i) the same variance for different groups, and (ii) the independent normal random variable , , for each level . Then the mean square within groups represents the mean square error (MSE), and becomes the point estimate of .

Under the null hypothesis the test statistic

has the F-distribution with degree of freedom. By we denote the critical point of -distribution with degree of freedom satisfying when X is the F-distributed random variable. In the hypothesis testing problem of one-way layout we reject with significance level when the observed value satisfies . Or, equivalently we can compute the p-value and reject when .